多重积分的应用,重谢!
来源:学生作业帮 编辑:百度作业网作业帮 分类:数学作业 时间:2024/07/22 10:41:52
多重积分的应用,重谢!
![](http://img.wesiedu.com/upload/d/d4/dd4dc0152b25f46e7f0a2e43b867041c.jpg)
![](http://img.wesiedu.com/upload/d/d4/dd4dc0152b25f46e7f0a2e43b867041c.jpg)
![多重积分的应用,重谢!](/uploads/image/z/19098437-5-7.jpg?t=%E5%A4%9A%E9%87%8D%E7%A7%AF%E5%88%86%E7%9A%84%E5%BA%94%E7%94%A8%2C%E9%87%8D%E8%B0%A2%21)
D: x^2+y^2=ax, 即 r=acost, 0≤r≤acost, -π/2≤t≤π/2.
曲面 z=±√(a^2-x^2-y^2), 则
dS=√[1+(z')^2+(z')^2]dxdy=adxdy/√(a^2-x^2-y^2)
=ardrdt/√(a^2-r^2),
由对称性,曲面有相同的2块,得
S = 2∫∫√[1+(z')^2+(z')^2]dxdy
= 2∫∫adxdy/√(a^2-x^2-y^2)
因D关于x轴对称,积分函数是y的偶函数,则
S = 4∫dt∫ardr/√(a^2-r^2)
= 2a∫dt∫[-d(a^2-r^2)]/√(a^2-r^2)
= 2a∫dt[-2√(a^2-r^2)]
= 4a^2∫(1-sint)dt
=4a^2[t+cost] = 4a^2(π/2-1) = 2(π-2)a^2.
曲面 z=±√(a^2-x^2-y^2), 则
dS=√[1+(z')^2+(z')^2]dxdy=adxdy/√(a^2-x^2-y^2)
=ardrdt/√(a^2-r^2),
由对称性,曲面有相同的2块,得
S = 2∫∫√[1+(z')^2+(z')^2]dxdy
= 2∫∫adxdy/√(a^2-x^2-y^2)
因D关于x轴对称,积分函数是y的偶函数,则
S = 4∫dt∫ardr/√(a^2-r^2)
= 2a∫dt∫[-d(a^2-r^2)]/√(a^2-r^2)
= 2a∫dt[-2√(a^2-r^2)]
= 4a^2∫(1-sint)dt
=4a^2[t+cost] = 4a^2(π/2-1) = 2(π-2)a^2.